Example 1: Identify the following random variables as discrete or continuous: a) Weight of a package. b) Number of students in a first-grade classroom

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1 Section 5-1 Probability Distributions I. Random Variables A variable x is a if the value that it assumes, corresponding to the of an experiment, is a or event. A random variable is if it potentially can take on any value on an. A random variable is if it can potentially assume only a or number of values. Identify the following random variables as discrete or continuous: a) Weight of a package b) Number of students in a first-grade classroom c) Age of a pediatric patient Example 2: An experiment consists of tossing fair coins. Let x be the random variable that is the number of in the five tosses. Is x discrete or continuous? List the sample space of values for x. Example 3: Let x be the random variable that gives the amount of time it takes a person to drive to work. Is x discrete or continuous? List the sample space of values of x.

2 II. Discrete Probability Distribution A or that lists the possible values of a discrete random variable and their probabilities is called the of the random variable. The probabilities that a customer will purchase 0, 1, 2, or 3 books are 0.42, 0.30, 0.15, and 0.10, respectively. (a) Construct a probability distribution for the data. (b) Draw a graph for the distribution. Example 2: Consider families with children. Let x be the number of in a family. Find the probability distribution for x and construct a graph for the probability distribution.

3 III. Requirements for a Discrete Probability Distribution A random variable x has this probability distribution: x P(x).2.3.1? (a) Find P(x = 3) (b) What is the probability that x is greater than 0? (c) What value of x is most likely to occur? (d) What is P(x = 8)? Example 2: Determine whether the distribution represents a probability distribution. If it does not, state why. x P(x)

4 Ch5-2 Mean, Variance Standard Deviation, and Expectation I. The of a discrete probability distribution II. The of a discrete probability distribution Definition formula for 2 The standard deviation is the square root of the variance. 2 Var For the probability distribution given below, find (a) the mean, (b) the variance, and (c) the standard deviation. Also (d) construct a graph for the probability distribution and describe the shape of the distribution. x P(x) 0 1/10 1 4/10 2 3/10 3 2/10

5 II. Expected Value For a discrete random variable x, the of x is the of the random variable x. E(x)= (Ref: General Statistics by Chase/Bown, 4 th Ed.) A high school class decides to raise some money by conducting a raffle. The students plan to sell 2000 tickets at $1 apiece. They will give one prize of $100, two prizes of $50, and three prizes of $25. If you plan to purchase one ticket, what are your expected net winnings? Example 2: (Ref: Elementary Statistics by Triola, 9 th edition) In New Jersey s Pick 4 lottery game, you pay 50 cents to select a sequence of four digits, such as If you win by selecting the same sequence of four digits that are drawn, you collect $2,788. (a) How many different selections are possible? (b) What is the probability of winning? (c) If you win, what is your net profit? (d) Find the expected value.

6 ch5-3 The Binomial Probability Distribution I. Binomial Experiment 1. Each has possible outcomes. 2. Perform an experiment a number of times (e.g. n is fixed). 3. The of each trial must be of each other. 4. The probability of a must remain the for each trial. The of a binomial experiment and the corresponding of these outcomes produce a binomial distribution. (Ref: Exploring Statistics by Kitchens, 2 nd ed.) Which of the following are binomial random variables? (a) The number of successful heart transplants out of five patients. (b) The length of a prison term for possession of marijuana (c) The name of each student in Math 227 (d) The number of approved food stamp recipients out of 50 applications II. Binomial Formula and Binomial Table A binomial experiment consists of identical trials with probability of success on each trial. The probability of successes in trials is P(x) = (a) Find 8C 3 (b) Find 12C 7 Example 2: Consider a binomial experiment with n = 15, p =.3, and x = number of success. Use the Binomial Formula for P(x) to find the probability that (a) x = 11 (b) x is less than 2

7 Example 3: It was found that of American victims of health care fraud are senior citizens. If 10 victims are randomly selected, find the probability that exactly 3 are senior citizens. Example 4: Consider a binomial experiment with,, and x = number of success. Use the Binomial Table to find the probability that (a) x is greater than 5 but less than 8 (b) x is greater than 7 (c) x equals 6 III. Mean and Variance of a Binomial Distribution For a binomial experiment consisting of n trials with the probability of success p, the or of x is For a binomial experiment consisting of n trials with the probability of success p, the of x is 2 Standard deviation = 2

8 Assume that 60% of a college s student loan applications are approved. Ten applications are chosen at random. (a) What is the probability that eight or more are approved? (b) How many applications are expected to be approved? (Find the mean of the number approved out of ten applications.) (c) What is the standard deviation of the number approved out of ten applications? Read Ch5 Discrete Probability Distributions P.266 Exercises 5-1: 1-5 all, 7, 9, all, 19, 21, 25, 27 P.275 Exercises 5-2: 1, 3, 5, 7,9, 11, 14, 15, 18 P.285 Exercises 5-3: 1, 2, 3, 4, 5, 7, 13, 14, 17, 21, 25 P. 292 Review Exercises: 1-9 all.

Probability Distributions. Definitions Discrete vs. Continuous Mean and Standard Deviation TI 83/84 Calculator Binomial Distribution

Probability Distributions. Definitions Discrete vs. Continuous Mean and Standard Deviation TI 83/84 Calculator Binomial Distribution Probability Distributions Definitions Discrete vs. Continuous Mean and Standard Deviation TI 83/84 Calculator Binomial Distribution Definitions Random Variable: a variable that has a single numerical value